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☻ 1.0 Axial Forces 2.0 Bending of Beams Now we consider the elastic deformation of beams (bars) under bending loads. M M www.engineering.auckland.ac.nz/mechanical/MechEng242 MECHENG242 Mechanics of Materials Bending of Beams Application to a Bar Normal Force: Fn Fn Bending Moment: S.B. Mt Mt Shear Force: Ft Ft Torque or Twisting Moment: K.J. Mn Mn MECHENG242 Mechanics of Materials Bending of Beams Examples of Devices under Bending Loading: Atrium Structure Excavator Yacht Car Chassis MECHENG242 Mechanics of Materials Bending of Beams 2.0 Bending of Beams 2.1 Revision – Bending Moments (Refer: B,C & A – Sec’s 6.1,6.2) 2.2 Stresses in Beams (Refer: B,C & A –Sec’s 6.3-6.6) sx sx P x Mxz 2.3 Combined Bending and Axial Loading P1 Mxz (Refer: B,C & A – Sec’s 6.11, 6.12) P2 2.4 Deflections in Beams 2.5 Buckling (Refer: B,C & A –Sec’s 7.1-7.4) (Refer: B,C & A –Sec’s 10.1, 10.2) MECHENG242 Mechanics of Materials Bending of Beams 2.1 Revision – Bending Moments RECALL… (Refer: B, C & A – Chapter 6) Last year Jason Ingham introduced Shear Force and Bending Moment Diagrams. 12 kN 3m 3m Q (SFD) 0 (BMD) M 0 MECHENG242 Mechanics of Materials Bending of Beams Consider the simply supported beam below: (Refer: B, C&A – Sections 1.14, 1.15, 1.16, 6.1) y Radius of Curvature, R P x B A Mxz RAy Mxz Deflected Shape Mxz MECHENG242 Mechanics of Materials Mxz RBy What stresses are generated within, due to bending? Bending of Beams P A W B u RAy Recall: Axial Deformation Load (W) Mxz Mxz RBy Bending Bending Moment (Mxz) Axial Stiffness Extension (u) MECHENG242 Mechanics of Materials Flexural Stiffness Curvature (1/R) Bending of Beams Axial Stress Due to Bending: y x Mxz=Bending Moment Mxz sx (Compression) Mxz sx=0 Beam sx (Tension) Unlike stress generated by axial loads, due to bending: sx is NOT UNIFORM through the section depth sx DEPENDS ON: (i) Bending Moment, Mxz (ii) Geometry of Cross-section MECHENG242 Mechanics of Materials Bending of Beams Sign Conventions: Qxy=Shear Force y Mxz=Bending Moment Mxz Mxz Qxy Qxy x -ve sx +ve sx +VE (POSITIVE) “Happy” Beam is +VE MECHENG242 Mechanics of Materials “Sad” Beam is -VE Bending of Beams Example 1: Bending Moment Diagrams Mxz=P.L P A y x B L RAy=P P.L Mxz P Mxz Qxy Mxz Qxy Q & M are POSITIVE MECHENG242 Mechanics of Materials Qxy x P Mxz Qxy F 0 ; M 0 ; y z Qxy P Mxz PL x Bending of Beams Q xy P; Mxz PL x P L y P.L Mxz B A x Qxy x P P +ve Qxy Mxz Shear Force Diagram 0 (SFD) 0 Bending Moment Diagram -P.L -ve (BMD) To find sx and deflections, need to know Mxz. MECHENG242 Mechanics of Materials Bending of Beams Example 2: Macaulay’s Notation a A R Ay z b x B R By x Pa a b P a M y C P b a b P b a b P Mxz A Qxy 0 ; Mxz P x a Mxz Where Pb x P x a a b x a MECHENG242 Mechanics of Materials Pb x 0 a b can only be +VE or ZERO. Bending of Beams P a A C Pb a b b x B Pa a b x (i) When x a : Mxz x a: (ii) When BMD: y Mxz Pab a b Mxz Pb x P x a a b 0 1 Pb x P x a a b 2 Eq. 1 Eq. 2 +ve 0 A MECHENG242 Mechanics of Materials C B Bending of Beams Distributed Load w per unit length Example 3: Distributed Load y x wL2 Mxz= 2 A B L x RAy=wL wL2 2 Mxz wx Qxy wL Mxz Qxy F 0 ; M 0 ; y wL wx Qxy 0 Qxy wL x w L2 x w Lx w x 0 Mxz z 2 2 MECHENG242 Mechanics of Materials Bending of Beams Qxy wL x Mxz w x2 w L2 w Lx 2 2 Mxz BMD: L 0 -ve x -wL2 2 MECHENG242 Mechanics of Materials w L2 2 @ x 0; Mxz @ x L; Mxz 0 L @x ; 2 Mxz w L2 8 Bending of Beams Summary – Is anything Necessary for Revision Generating Bending Moment Diagrams is a key skill you must revise. From these we will determine: • Stress Distributions within beams, • and the resulting Deflections Apart from the revision problems on Sheet 4, you might try these sources: • B, C & A Worked Examples, pg 126-132 Problems, 6.1 to 6.8, pg 173 • Jason Ingham’s problem sheets: www.engineering.auckland.ac.nz/mechanical/EngGen121 MECHENG242 Mechanics of Materials Bending of Beams